How can I use qr() in MATLAB to compute LQ - Decomposition?

Question

I want to compute this:

$$\begin{bmatrix} U\\ Y \end{bmatrix} = \begin{bmatrix} L_{11} &0 \\ L_{21} & L_{22} \end{bmatrix}\begin{bmatrix} Q_1\\ Q_2 \end{bmatrix}$$

Is this matlab command right then?

 >> L = tril(qr([U;Y]))

The MATLAB command tril is lower-traingle function. Is this right way to compute the LQ - Decomposition?

The reason why I asking this simple question, is because a lot of books talking about LQ - Decomposition but not explaining how it's done.

Answer 1

Here is the answer - In MATLAB way!

Assume that we have matrix U and Y.

>> U = magic(5)
U =

   17   24    1    8   15
   23    5    7   14   16
    4    6   13   20   22
   10   12   19   21    3
   11   18   25    2    9

>> Y = 5*magic(5)
Y =

   85   120     5    40    75
  115    25    35    70    80
   20    30    65   100   110
   50    60    95   105    15
   55    90   125    10    45

Then we want to solve this:

$$\begin{bmatrix} U\\ Y \end{bmatrix} = \begin{bmatrix} L_{11} &0 \\ L_{21} & L_{22} \end{bmatrix}\begin{bmatrix} Q_1\\ Q_2 \end{bmatrix}$$

Then we do this:

>> [Q, L] = qr([U;Y]')
Q =

  -0.500216   0.509947   0.693606  -0.067606  -0.063859
  -0.706188  -0.654162  -0.056572  -0.263444  -0.027712
  -0.029424   0.312464  -0.367193  -0.546166  -0.684377
  -0.235396   0.398863  -0.432121  -0.339759   0.695221
  -0.441367   0.235160  -0.440626   0.715746  -0.208446

L =

   -33.98529   -25.59931   -21.03851   -20.30290   -23.39247  -169.92645  -127.99655  -105.19257  -101.51451  -116.96236
     0.00000    19.99188    15.32765    12.26797     4.56029     0.00000    99.95941    76.63825    61.33983    22.80146
     0.00000     0.00000   -20.67472   -11.11590    -7.39834     0.00000    -0.00000  -103.37360   -55.57948   -36.99169
     0.00000     0.00000     0.00000   -19.20224   -13.37762     0.00000    -0.00000     0.00000   -96.01121   -66.88809
     0.00000     0.00000     0.00000     0.00000   -18.79627    -0.00000    -0.00000     0.00000    -0.00000   -93.98136

>>

We rewrite variable L to this:

>> L = L'
L =

   -33.98529     0.00000     0.00000     0.00000     0.00000
   -25.59931    19.99188     0.00000     0.00000     0.00000
   -21.03851    15.32765   -20.67472     0.00000     0.00000
   -20.30290    12.26797   -11.11590   -19.20224     0.00000
   -23.39247     4.56029    -7.39834   -13.37762   -18.79627
  -169.92645     0.00000     0.00000     0.00000    -0.00000
  -127.99655    99.95941    -0.00000    -0.00000    -0.00000
  -105.19257    76.63825  -103.37360     0.00000     0.00000
  -101.51451    61.33983   -55.57948   -96.01121    -0.00000
  -116.96236    22.80146   -36.99169   -66.88809   -93.98136

And we rewrite variable Q to this:

>> Q = Q'
Q =

  -0.500216  -0.706188  -0.029424  -0.235396  -0.441367
   0.509947  -0.654162   0.312464   0.398863   0.235160
   0.693606  -0.056572  -0.367193  -0.432121  -0.440626
  -0.067606  -0.263444  -0.546166  -0.339759   0.715746
  -0.063859  -0.027712  -0.684377   0.695221  -0.208446

Now we check if [U;Y] = L*Q

>> [U;Y]
ans =

    17    24     1     8    15
    23     5     7    14    16
     4     6    13    20    22
    10    12    19    21     3
    11    18    25     2     9
    85   120     5    40    75
   115    25    35    70    80
    20    30    65   100   110
    50    60    95   105    15
    55    90   125    10    45

>> L*Q
ans =

    17.0000    24.0000     1.0000     8.0000    15.0000
    23.0000     5.0000     7.0000    14.0000    16.0000
     4.0000     6.0000    13.0000    20.0000    22.0000
    10.0000    12.0000    19.0000    21.0000     3.0000
    11.0000    18.0000    25.0000     2.0000     9.0000
    85.0000   120.0000     5.0000    40.0000    75.0000
   115.0000    25.0000    35.0000    70.0000    80.0000
    20.0000    30.0000    65.0000   100.0000   110.0000
    50.0000    60.0000    95.0000   105.0000    15.0000
    55.0000    90.0000   125.0000    10.0000    45.0000

>>

Yes it is! Please, correct me if I'm wrong!

Answer 2

Suppose you want a $LQ$ factorization of a matrix $A$. You then do a $QR$ factorization of $A^T$, i.e., $A^T=UR$, where $U$ is orthogonal and $R$ is upper triangular. Then $A = R^T U^T$ and $L = R^T$ is lower triangular, while $Q = U^T$ is orthogonal.